This issue is likely related to Sugar not yet supporting OAuth 2, which is now required for Google's 2-step Verification. Sugar currently uses ClientLogin which has been deprecated by Google. However, we are scheduled to support OAuth 2 with the release of 7.6.1.0, which is our next planned 7.6 release. I would recommend that you follow our Releases category here in the community to be notified when this version is released, at which point you can upgrade to 7.6.1.0. After that, you should be all set!

Is there any planning for preventing the disfunction of core features of SugarCRM because of usage of deprecated APIs in the future? ClientLogin was deprecated in April 2014, so there was a year to implement the new client api...

Hey mwillert, Absolutely. We have dedicated teams now keeping an eye out for API deprecation and changes in order to not disrupt our customers business processes. Thank you for raising your concern. Please know that we are on top of this moving forward.

You definitely have this right. One caveat, be sure to create an application for Drive or Contacts if you are trying to integrate Google Docs or Google Contacts with Sugar. I have seen people create the keys in Google but not create the applications and it causes issues. Admin > Connectors > Set Connector Properties > Google should explain the basics.